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Opportunities for the direct manipulation of a phase-driven Andreev spin qubit

Yoan Fauvel

Published Apr 9, 2024• 0 citations
Publisher

Abstract

Opportunités de manipulation directe d'un qubit de spin d'Andreev piloté en phase The research presented in this thesis delves into a theoretical exploration of Andreev bound states (ABS) hosted in Josephson junctions with spin-orbit coupling (SOC). ABS are fermionic bound states that appear at the junction between two superconducting electrodes forming a Josephson junction. These bound states are at the core of the microscopic description of the Josephson effect and provide an explanation for the transfer of Cooper pairs from one superconducting electrode to the other. Crucially, when the superconducting phase difference between the two electrodes is non-zero, a supercurrent can flow in the junction, depending on the occupancy of the ABS. The presence of this supercurrent facilitates the coupling between a microwave resonator and a junction due to its sensibility to the electromagnetic field. This coupling then allows for the detection and manipulation of ABS.Furthermore, in the presence of SOC along with a superconducting phase bias, the spin degeneracy of ABS can be lifted. Therefore such a junction provides a unique opportunity to realize a special kind of spin qubit known as Andreev spin qubit (ASQ). The qubit operation can be performed via an AC modulation of an electrostatic gate or magnetic flux, thanks to the sensitivity of the ABS to the electric potential or the phase difference, respectively. So far, the coherent manipulation of such qubits has been realized in gate-driven experiments as well as in flux-driven experiments with the use of additional Andreev levels. However, the direct transition between the two ABS forming the ASQ has remained out of reach in flux-driven experiments. In this thesis, we investigate two distinct types of Josephson junctions with SOC with the aim of estimating the amplitude of the matrix elements of the current operator between two ABS forming an ASQ. These elements characterize the coupling strength between the qubit and an external flux drive, indicating which transitions between ABS are within reach. Our first project focuses on a superconductor-normal-superconductor junction where the normal region consists of a nanowire with Rashba SOC. As a minimal model, we consider a generic scatterer located along the nanowire. Unless this scatterer possesses additional spatial (mirror) symmetries, it is responsible for inducing spin-flip transmission probability. Using the scattering formalism, we derive the energy spectrum of ABS and assess how SOC influences the spin-splitting between opposite spin states. We then obtain analytical expressions for the matrix elements of the current operator. Notably, our study reveals that SOC allows one to have finite elements between opposite spin states. In a second project, the system we study consists of a superconductor-double quantum dots-superconductor junction. A quantum dot consists of a tiny region defined by electrostatic gates or impurities. Due to its small size, Coulomb repulsion can be significant. However, in the simplest case of a quantum dot with a single level, even in the presence of SOC in the coupling between the dot and the leads, spin degeneracy of ABS is still preserved. However, recent experiments showed that additional levels allow to lift the spin degeneracy. Here, we introduce a second quantum dot which provides an additional level and offers more control on each level. By using an effective model, we identify the minimal ingredients that are necessary to lift the spin degeneracy. Our results reveal that this splitting is achievable through spin-dependent couplings with the leads and is reduced by the Coulomb repulsion. Additionally, the presence of a finite coupling between the two dots, combined with SOC in the couplings with the leads, allows one to have finite matrix elements of the current operator between opposite spin states, which we obtain numerically. In short, we present in this thesis two platforms for the realization of phase-driven ASQ.

DOI
10.70675/238023ebz1c15z4101za90aza1e61aefc986

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Opportunités de manipulation directe d'un qubit de spin d'Andreev piloté en phase The research presented in this thesis delves into a theoretical exploration of Andreev bound states (ABS) hosted in Josephson junctions with spin-orbit coupling (SOC). ABS are fermionic bound states that appear at the junction between two superconducting electrodes forming a Josephson junction. These bound states are at the core of the microscopic description of the Josephson effect and provide an explanation for the transfer of Cooper pairs from one superconducting electrode to the other. Crucially, when the superconducting phase difference between the two electrodes is non-zero, a supercurrent can flow in the junction, depending on the occupancy of the ABS. The presence of this supercurrent facilitates the coupling between a microwave resonator and a junction due to its sensibility to the electromagnetic field. This coupling then allows for the detection and manipulation of ABS.Furthermore, in the presence of SOC along with a superconducting phase bias, the spin degeneracy of ABS can be lifted. Therefore such a junction provides a unique opportunity to realize a special kind of spin qubit known as Andreev spin qubit (ASQ). The qubit operation can be performed via an AC modulation of an electrostatic gate or magnetic flux, thanks to the sensitivity of the ABS to the electric potential or the phase difference, respectively. So far, the coherent manipulation of such qubits has been realized in gate-driven experiments as well as in flux-driven experiments with the use of additional Andreev levels. However, the direct transition between the two ABS forming the ASQ has remained out of reach in flux-driven experiments. In this thesis, we investigate two distinct types of Josephson junctions with SOC with the aim of estimating the amplitude of the matrix elements of the current operator between two ABS forming an ASQ. These elements characterize the coupling strength between the qubit and an external flux drive, indicating which transitions between ABS are within reach. Our first project focuses on a superconductor-normal-superconductor junction where the normal region consists of a nanowire with Rashba SOC. As a minimal model, we consider a generic scatterer located along the nanowire. Unless this scatterer possesses additional spatial (mirror) symmetries, it is responsible for inducing spin-flip transmission probability. Using the scattering formalism, we derive the energy spectrum of ABS and assess how SOC influences the spin-splitting between opposite spin states. We then obtain analytical expressions for the matrix elements of the current operator. Notably, our study reveals that SOC allows one to have finite elements between opposite spin states. In a second project, the system we study consists of a superconductor-double quantum dots-superconductor junction. A quantum dot consists of a tiny region defined by electrostatic gates or impurities. Due to its small size, Coulomb repulsion can be significant. However, in the simplest case of a quantum dot with a single level, even in the presence of SOC in the coupling between the dot and the leads, spin degeneracy of ABS is still preserved. However, recent experiments showed that additional levels allow to lift the spin degeneracy. Here, we introduce a second quantum dot which provides an additional level and offers more control on each level. By using an effective model, we identify the minimal ingredients that are necessary to lift the spin degeneracy. Our results reveal that this splitting is achievable through spin-dependent couplings with the leads and is reduced by the Coulomb repulsion. Additionally, the presence of a finite coupling between the two dots, combined with SOC in the couplings with the leads, allows one to have finite matrix elements of the current operator between opposite spin states, which we obtain numerically. In short, we present in this thesis two platforms for the realization of phase-driven ASQ.
Conventional tunnel Josephson junctions form the basic building blocks of superconducting electronics. Their operation is based on comparatively simple dynamics where the superconducting phases across the junctions is the only dynamical degree of freedom. The goal of the project on which this thesis is based, is to understand the physics and develop a consistent theoretical description of the quantum dynamics of unconventional Josephson structures: spin active junctions, junctions containing low energy quasiparticles, and junctions composed of ddd-wave superconductors. Spin active Josephson junctions involve the additional degree of freedom of quasiparticle spin. Here we theoretically investigate the spin-structure of the current carrying interface bound states - Andreev Bound States (ABS), and induced ground state spin polarization of a short (shorter than the superconducting coherence length) nanowire junction containing a localized magnetic moment. We show that temporal variation of the superconducting phase difference and magnetic moment induce rotations in spin polarized and unpolarized subspaces respectively, which is manifested by temporal oscillation of the Josephson current.The interlevel transitions obey a selection rule that forbids manipulations in a certain region ofexternal parameters and results from specific properties of Andreev bound states in magnetic contacts where, in a certain regime of phase bias and strength of magnetic scatterer, ABS belong to different eigenstates of the combined charge and parity inversion symmetry operator.A general framework is developed to include low energy quasiparticle states into the dynamics of Josephson junctions. Such states exist in e.g. constriction type junctions with high transparency channels or resonant states, as well as in junctions of unconventional superconductors. The existence of such states prevent the separation of energy scales which in conventional tunnel junctions freezes out the quasiparticle degrees of freedom. We show that the resonant interaction with such low energy states rather than the Josephson potential defines the nonlinear Josephson dynamics at small amplitudes.The framework is applied to high temperature Josephson junctions, whose unconventional order parameter admits ABS with energies in the middle of the superconducting gap - Mid Gap States (MGS). In addition to the non-linear dynamics induced by the MGS, we identify a reentrance effect in the transition between thermal activation and macroscopic quantum tunneling associated with the MGS, and connect this phenomenon to experimental observations.
We propose a variant of Andreev spin qubits (ASQs) defined in planar Josephson junctions based on spin-orbit coupled two-dimensional electron gases (2DEGs) in a weak out-of-plane magnetic field. The magnetic field induces a linear phase gradient across the junction, generating Josephson vortices that can host low-energy Andreev bound states (ABSs). We show that, in certain parameter regimes, the combined effect of the phase gradient and spin-orbit coupling stabilizes an odd-fermion parity ground state, where a single Josephson vortex binds a spinful low-energy degree of freedom that is energetically separated from the other ABSs. This low-energy degree of freedom can be exploited to define a special type of ASQ, which we dub the vortex spin qubit (VSQ). We show that single-qubit gates for VSQs can be performed via flux driving, while readout can be achieved by adapting standard circuit quantum electrodynamics (cQED) techniques developed for conventional ASQs. We further outline how an entangling two-qubit gate can be performed using an ac current drive. We argue that VSQs offer prospects for a substantial reduction in device complexity and hardware overhead compared to conventional ASQ implementations, while preserving key advantages such as supercurrent-based readout, single-qubit gates, and long-range two-qubit gates.
In this thesis we present theoretical investigations of the effects of Andreev bound states on the current transport in superconducting interferometers. We also investigate the slow dynamics of the Andreev states in a superconducting point contact, and the possible application as a quantum bit. We consider superconductor-normal metal-superconductor (SNS) and normal metalsuperconductor (NS) interferometers, where the contact region is a Y-shaped normal metal wave guide, and the two connection points to the same superconducting electrode can have different phases. The electric current in the interferometer is calculated as a function of the applied voltage and the phase difference . Andreev reflection in SNS and NS interferometers incorporates two features: interference in the arms of the Y-shaped normal region, and interplay with Andreev resonances. The latter feature yields rich phase dependent current structures in the subgap voltage region. The interference effect leads to a suppression of the current structures at . We investigate the effects on the Josephson current in NS interferometers due to current injection from the normal electrode. The two main effects of the nonequilibrium situation are: nonequilibrium population of the Andreev levels, which can result in enhancement, suppression, or even sign reversal of the Josephson current, and an anomalous interference Josephson effect, which gives rise to a long range Josephson effect, increasing with the voltage up to the superconducting gap . The two Andreev states in a superconducting quantum point contact can be accessed for manipulation and measurement by embedding the point contact in a superconducting loop. We calculate an effective Hamiltonian for the slow dynamics of the Andreev two-level system in the ring. Furthermore, we discuss methods of manipulation of the Andreev levels, and coupling of qubits. The state of the Andreev two-level system can be read out by monitoring the macroscopic quantum tunneling in a current biased Josephson junction, which is embedded in the superconducting ring of the qubit. We discuss the effects on the qubit, the readout scheme and the signal-to-noise ratio. THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Coherent processes in Superconducting quantum interferometers and qubits
We investigate the superconducting Anderson impurity model for interacting quantum dot Josephson junctions with spin-orbit coupling and a term accounting for tunnelling through higher-energy orbitals. These elements establish the conditions required for spin polarization in the absence of external magnetic field at finite superconducting phase bias. This Hamiltonian has been previously used to model the Andreev spin qubit, where quantum information is encoded in spinful odd-parity subgap states. Here we instead analyse the even-parity sector, i.e., the Andreev pair qubit based on Andreev bound states (ABS). The model is solved using the zero-bandwidth approximation and the numerical renormalization group, with further insight from variational calculations. Electron-electron interaction admixes single-occupancy Yu-Shiba-Rusinov (YSR) components into the ABS states, thereby strongly enhancing spin transitions in the presence of spin-orbit coupling. The ABS states can thus become sensitive to local magnetic field fluctuations, which has implications for decoherence in Andreev pair qubits. For strong interaction U U , especially in the cross-over region between the ABS and YSR regimes for U \sim 2\Delta U ∼ 2 Δ , charge, spin, and inductive transitions can all become strong, offering avenues for spin control and quantum transduction.
Superconducting spin qubits, also known as Andreev spin qubits, promise to combine the benefits of superconducting qubits and spin qubits defined in quantum dots. While most approaches to control these qubits rely on controlling the spin degree of freedom via the supercurrent, superconducting spin qubits can also be coupled to each other via the superconductor to implement two-qubit quantum gates. We theoretically investigate the interaction between superconducting spin qubits in the weak tunneling regime and concentrate on the effect of spin-orbit interaction (SOI), which can be large in semiconductor-based quantum dots and thereby offers an additional tuning parameter for quantum gates. We find analytically that the effective interaction between two superconducting spin qubits consists of Ising, Heisenberg, and Dzyaloshinskii-Moriya interactions and can be tuned by the superconducting phase difference, the tunnel barrier strength, or the SOI parameters. The Josephson current becomes dependent on SOI and spin orientations. We demonstrate that this interaction can be used for fast controlled phase-flip gates with a fidelity >99.99%. We propose a scalable network of superconducting spin qubits which is suitable for implementing the surface code.
We theoretically study the dephasing of an Andreev spin qubit (ASQ) due to electric and magnetic noise. Using a tight-binding model, we calculate the Andreev states formed in a Josephson junction where the link is a semiconductor with strong spin-orbit interaction. As a result of both the spin-orbit interaction and induced superconductivity, the local charge and spin of these states vary as a function of externally controllable parameters: the phase difference between the superconducting leads, an applied magnetic field, and filling of the underlying semiconductor. Concomitantly, coupling to fluctuations of the electric or magnetic environment will vary, which informs the rate of dephasing. We qualitatively predict the dependence of dephasing on the nature of the environment, magnetic field, phase difference between the junction, and filling of the semiconductor. Comparing the simulated electric- and magnetic-noise-induced dephasing rate to experiment suggests that the dominant source of noise is magnetic. Moreover, by appropriately tuning these external parameters, we find sweet spots at which we predict an enhancement in ASQ coherence times.